Bibliography

P. A. M. Dirac. The Principles of Quantum Mechanics, 3d edition, Oxford, 1947.

P. A. M. Dirac. Principles of Quantum Mechanics. 3rd edition.

Dirac’s well-known book, which appeared in 1930, was the first fundamental monograph on quantum mechanics, setting out both the abstract foundations and the principal applications of the nonrelativistic theory, completed by its construction in 1927–28. One chapter was devoted to the relativistic theory of the electron, only just established by Dirac himself in 1928.

This book, despite its rigor and a certain abstractness, met with an enthusiastic reception among broad circles and was translated into the principal international languages. The Russian edition appeared in 1932. The second edition, published in 1935 and likewise translated into Russian, as Dirac wrote in the preface to it, differed from the first by somewhat lesser abstractness. The concept of state was applied in a three-dimensional nonrelativistic sense. Sections on quantum electrodynamics and the second quantization of Fermi particles were added.

The present, third edition of Dirac’s book, apart from various less significant changes in the text, has been almost completely rewritten and, according to the words of the preface, differs from the previous editions chiefly in the following three respects.

Quantum electrodynamics is presented with allowance for the Wentzel–Dirac lambda-process theory and the classical theory of the point electron, proposed by Dirac and developed especially by Sokolov and Bhabha. Let us recall that here, in the classical theory, the matter at issue is the derivation or justification of the equation of motion of a point electron in such a way that the familiar terms, depending on its radius, obtained by Lorentz, would fall away naturally. Dirac’s theory rejects the hypothesis of the field mass of the electron, which here simply disappears, leaving the question of the nature of the proper mass open for the time being. The terms with the reaction of radiation friction coincide exactly for Lorentz and Dirac. In this connection it should be noted that recently in quantum electrodynamics a careful analysis of various infinities has been carried out, and it has become clear that discarding infinite (in the contemporary imperfect formalism) proper inertial terms of the type of mass or moment of inertia leads to reasonable results. We have in mind the quantum theory of damping in the continuous spectrum by Sokolov–Heitler–Wilson and the recent work of Bethe and of the Oppenheimer group (Lewis, Epstein, and others) on the shift of levels in the hydrogen atom. Characteristic of the Dirac–Sokolov theory is the use of advanced potentials along with retarded ones.

As Dirac further shows, the classical relativistic, but nonquantum, equations of motion of a point electron in an electromagnetic field can be formulated in the form of Hamiltonian equations with Poisson—

with new brackets, if for the field potentials one takes the Wentzel, and not the Maxwell, values.

The Poisson brackets for the Wentzel potentials depend on a certain auxiliary time-like vector \(\Lambda\), which in the end tends to zero. Replacing Poisson brackets by commutation relations, we pass, as usual, to quantum theory.

It is not difficult to show that the vector \(\Lambda\) plays the role of the electron form factor or of an auxiliary cut-off multiplier in the right-hand side of the commutation rule. The field mass of the electron, due to the longitudinal part of the field, calculated with the aid of the Wentzel–Dirac potentials, or, in other words, with the aid of the new commutation rules, proves to be equal to zero.

Let us recall that, figuratively speaking, the \(\Lambda\)-process is to a considerable extent reducible to calculating the force acting on the electron at a certain moment of time as the average of the forces acting at a moment somewhat later and somewhat earlier, and moreover in the final formulas this shift \((\Lambda)\) tends to zero. However, the application of the limiting \(\Lambda\)-process does not remove the infinite field energy produced by a point particle, associated with its transverse part and having a purely quantum origin. Since this energy must be equated to the (transverse) proper mass, the latter proves to be infinite. Dirac sees no way out of this difficulty in quantum electrodynamics and thereby renounces his earlier proposal to eliminate the transverse infinite energy by the artificial introduction of negative energies for photons or negative probabilities (see his Dublin lectures).

Hardly any theorist will bewail the abandonment of this complicated and unfruitful hypothesis. It should not be forgotten that Dirac’s \(\Lambda\)-process, on the one hand, is arbitrary and ambiguous as a cut-off multiplier and, on the other hand, has led to no physical results. As a device merely for eliminating the longitudinal infinite mass, it is, truly, too costly a remedy!

In concluding the chapter on electrodynamics, Dirac points out the necessity, in calculating processes of scattering, emission, and absorption, of cutting off the divergent integrals associated with transverse waves at wavelengths of the order of the electron radius, at the same time emphasizing the unsatisfactory character of this procedure and hinting at the possibility of radical changes in the theory. The first part of Dirac’s remark is, however, surprising, since it is well known that quantum electrodynamics, contrary to the initial pessimism, has proved suitable also for many processes with wavelengths smaller than the electron radius, as became clear, for example, in the cascade theory of cosmic-ray showers.

Although the formalism of lambda-processes and Wentzel potentials was applied up to now only for discussing fundamental questions, and the entire exposition of Dirac’s quantum electrodynamics is far from real problems close to empiricism (the limits of cascade theory, the problem of level shifts, etc.), this chapter is nevertheless unquestionably interesting and provides occasion for an instructive discussion. Here, however, the more or less significant physical novelties of the book come to an end.

The second new point is connected with a modified exposition of the theory of systems of identical particles. Finally, the third new point consists in the use of a new symbolism, the so-called “dual” vectors: “bra” (bra) and “ket” (ket) (sic!). In an astonishing way Dirac himself places precisely this point first as the most important improvement of the new edition of his book. These terms, strange-sounding to the non-Saxon ear, are evidently halves of the word bracket, i.e. bracket. Therefore a Russian translation might sound like vectors of the type “bracket”

and of the type “ket” (or “c” and “bra”?). The motives that prompted Dirac to reject the customary terms \(\psi\) (“ket”) and \(\psi^*\) (“bra”) do not seem convincing to us, and the transcription of the type \(|\xi>\) (instead of \(\psi_\xi^*\)) and \(<\xi'|\xi''>=0\) for \(\xi'\ne\xi''\), instead of \(\int \psi_{\xi'}^*\psi_{\xi''}\,d\tau=0\) (the orthogonality theorem), in our opinion offers no special convenience. A certain economy in writing is not redeemed by the excessive abstractness, by the absence of visual clarity, and by the break with the traditions of mathematical and physical transcription.

Let us recall that, when Dirac introduced the German terminology eigen-value (proper values) instead of proper-value, certain protests were heard in the English literature. In any case, such theorems as the following (p. 31) are a real rebus for the uninitiated reader: “the eigen-values associated with eigen-kets are the same as the eigen-values associated with eigen-bras,” i.e., translating in Dirac’s spirit: eigen-values associated with eigen-kets coincide with eigen-values associated with eigen-bras. In the old “outmoded” language this means: “proper values associated with the vectors \(\psi_\xi\) coincide with the proper values associated with the conjugate vectors \(\psi_\xi^*\).” Or (p. 55) the equation
\[ <\lambda_1\lambda_2\cdots\lambda_n/L_1=\lambda_1><\lambda_1\lambda_2\cdots\lambda_n \]
“shows that each basic bra is an eigen-bra of \(L_1\), the value of the parameter \(\lambda_1\), being the eigen-value belonging to it” (the translation of which we leave to the reader). In what language is all this written, and is not the new terminology a kind of joke?

After this it is not surprising that the appearance of the wave function \(\psi\) on p. 79 is experienced almost as a meeting with a friend:
if
\[ <\xi'|P>=\psi(\xi'), \tag{60} \]
then
\[ |P>=|\psi(\xi)>. \]

With the aid of such terminology Dirac expounds anew the general theory of representations of linear operators in Chapters 2 and 3 of the book. Chapters 3 and 4 of the second edition on the theory of representations have now been merged into a single third chapter; the content and headings of the remaining chapters have remained almost without change. We note still other, rather successful terms: bosons and fermions (particles or systems obeying Bose or Fermi statistics).

However important the justification, axiomatics, and working out of terminology and transcription in quantum theory may be, we cannot fail to regard in the excessive attention of Dirac to these problems a sign of his withdrawal from the front-line tasks of our science into a rather deep rear. There is nothing in the book about photon annihilation, about the nucleus, nuclear forces, cosmic rays, or the meson, notwithstanding the fact that two of its chapters (11 and 12) deal with problems of the relativistic theory of the electron and quantum electrodynamics, and in the preceding chapters many applications of nonrelativistic quantum mechanics are given.

It seems to us that this departure of Dirac from the current problems of our science is not accidental, but is conditioned, to a certain extent, by his philosophical positions of the “Cambridge school,” inclined toward formalism and even toward scholasticism in its extreme manifestations (for example, the derivation of world constants by Eddington) after the exhaustion of the healthy empirical basis by one author or another. Of course, one cannot demand of a book devoted to the foundations of a theory an exhaustive exposition of applications, but all our science has advanced far forward since 1930.

The excellent exposition of Dirac’s famous quantum equation in Chapter 11 has undergone almost no changes in 18 years. As before, Dirac does not give here a general formula for the transformation of \(\psi\)-functions under a Lorentz transformation, limiting himself to separate cases, po-

as before, it is confined to analysis of the equation written only in polar coordinates alongside Cartesian ones.

Dirac has now abandoned the previous unclear treatment of the electron’s electric moment, and now in the exposition (pp. 264–265) there appears only the kinematic magnetic moment at low velocities

\[ \vec{\mu}_e = \frac{eh}{4\pi mc}\,\vec{\sigma}. \]

Since, however, the electric moment necessarily arises alongside the magnetic one, forming together with the latter the tensor of moments noted by Ya. I. Frenkel back in the classical theory of the electron, this new exposition of the question by Dirac cannot be regarded as sufficiently complete.

In our review we have dwelt chiefly on certain characteristic novelties of Dirac’s book, perhaps expressing with excessive emphasis a certain disappointment of our own. On the other hand, it should be stressed that the third edition contains an enormous wealth of valuable material, a large part of which has passed over from the first two; among these, first of all, one should note, for example, the brilliant introduction to the principle of superposition in § 1, the treatment of Poisson brackets (Ch. 4), the theory of the oscillator, the theory of angular momenta and the treatment of hydrogen in Ch. 6, Chapter 8, devoted to collisions and containing a beautiful formulation expressing the variational principle, the theory of radiation in Ch. 10, the relativistic theory of the electron in Ch. 11, and so on and so forth.

There is no doubt that the translation of the third edition of Dirac’s fundamental book, the first editions of which in Russian have long since been sold out, will serve fruitful discussions and will promote the further, more profound study of quantum mechanics in our country.

D. Ivanenko